Matching and Weighted P2P2-Packing: Algorithms and Kernels

Parameterized algorithms and kernelization algorithms are presented for the weighted P2P2-Packing problem, which is a generalization of the famous Graph Matching problem. The parameterized algorithms are based on the following new techniques and observations: (1) new study on structure relationship between graph matchings in general graphs and P2P2-packings in bipartite graphs; (2) an effective graph bi-partitioning algorithm; and (3) a polynomial-time algorithm for a constrained weighted P2P2-Packing problem in bipartite graphs. The kernelization algorithms are based on the following new techniques: (1) the application of graph matching in kernelization; (2) a crown reduction structure for weighted problems. These techniques lead to randomized and deterministic parameterized algorithms that significantly improve the previous best upper bounds for the problem for both weighted and unweighted versions. For the kernelization algorithm, by using a weighted version of crown reduction, a kernel of size O(k2)O(k2) is presented, where k is the given parameter of the problem.